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Chapter 2: Problem 42

Use a spreadsheet to calculate the specified term of each recursively defined sequence. If \(a_{n+1}=a_{n}+\frac{1}{2}\) and \(a_{0}=1\) find \(a_{12}\).

Short Answer

Expert verified
The 12th term, \(a_{12}\), is 7.

Step by step solution

01

Understand the recursive formula

The given sequence is defined recursively, where each term is computed based on the previous term. The formula is \(a_{n+1} = a_{n} + \frac{1}{2}\), and the initial term is \(a_0 = 1\).
02

Set up the spreadsheet formula

In the first cell (let's use cell A1), input the initial term, which is 1 for \(a_0\). In the next cell below it (A2), input the formula that adds \(\frac{1}{2}\) to the above cell's value. The formula for A2 will be "=A1 + 0.5".
03

Use the formula recursively

Use the spreadsheet's drag-down feature (autofill) to extend the formula for enough rows to reach \(a_{12}\). This will automatically apply the formula to each subsequent cell, effectively calculating the sequence recursively.
04

Identify \(a_{12}\) in the spreadsheet

After dragging down the formula to the 13th row (which corresponds to \(a_{12}\)), observe the value in that cell. This cell calculates the 12th term based on the recursive application of the formula.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spreadsheet Calculation
Using a spreadsheet for calculations is a wonderful way to handle sequences, especially when they are defined recursively. Students often find spreadsheets like Excel or Google Sheets ideal for this task because they can automate repetitive calculations.

Here鈥檚 how you can leverage spreadsheets for recursive sequences:
  • Initial Setup: Start by placing your initial term in the first cell. In this exercise, the initial term is 1, so you enter '1' in cell A1.
  • Formula Input: In the next cell (A2), input the formula that calculates the next term by referencing the previous cell and applying the recursive step. For this exercise, you'd input "=A1 + 0.5."
  • Drag Down: Use the drag-down (autofill) feature of the spreadsheet to copy the formula down a column until you've reached the term you want to determine. For instance, dragging from cell A2 to A13 gives us up to the 12th term.
This method not only saves time but also minimizes errors from manual calculations. As a result, spreadsheets become powerful tools in learning and confirming your understanding of sequences.
Recursive Formula
A recursive formula is a rule or equation that defines each term in a sequence using one or more of the previous terms.

In this particular exercise, the recursive formula is given as: \[ a_{n+1} = a_{n} + \frac{1}{2} \]This indicates that each subsequent term is calculated by adding 0.5 to the previous term.
  • Simplicity: This formula is straightforward because every new term just requires a simple addition to the prior term.
  • Importance: Recursive formulas are crucial because they allow you to build sequences incrementally, providing a piece-by-piece construction of each term.
  • Application: These formulas are used in various mathematical and real-world applications, especially where sequences naturally build on earlier values.
Understanding the essence of the recursive formula helps in grasping how sequences develop and how each term relates back to its predecessors.
Initial Term
The initial term is an essential part of a recursively defined sequence. It serves as the starting point from which all other terms are generated.For the given exercise, the initial term is:\[ a_0 = 1 \]Here's why the initial term is key:
  • Foundation: The entire sequence depends on this value. Without it, the recursive process cannot begin.
  • Unique Identification: The initial term helps in uniquely identifying a sequence, as changing it results in a completely different sequence outcome.
  • Calculation Starting Point: In spreadsheets, the initial term is the first value you input, ensuring the accuracy of all subsequent terms through the recursive formula.
Understanding the initial term is crucial for predicting how sequences unfold and ensures that students are clear on where and how the sequence begins in practical applications.

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Most popular questions from this chapter

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty} \frac{n+2^{-n}}{n} $$

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Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=3.1, x_{0}=0\)

Investigate the advantage of dimensionless variables. A population obeys the discrete logistic equation: $$ N_{t+1}=R_{0} \cdot N_{t}-b N_{t}^{2} $$ Find the possible fixed points of the population size (one fixed point will depend on the unknown parameters \(R_{0}\) and \(b\) ).

Model painkillers that are absorbed into the blood from a slow release pill. Ourmathematical model for the amount, \(a_{t}\), of drug in the blood t hours after the pill is taken must include the amount absorbed from the pill each hour. Our model starts with the word equation. $$ \begin{array}{c} a_{t+1}=a_{t}+\begin{array}{l} \text { amount absorbed } \\ \text { from the pill } \end{array}-\begin{array}{l} \text { amount eliminated } \\ \text { from the blood } \end{array} \end{array} $$ Assume the amount absorbed from the pill between time \(t\) and time \(t+1\) is \(20 \cdot(0.2)^{t}\). (a) The drug has first order elimination kinetics. \(40 \%\) of the drug is eliminated from the blood each hour. Write down the recursion relation for \(a_{t+1}\) in terms of \(a_{t}\) (b) Assuming that \(a_{0}=0\), meaning that no drug is present in the blood initially, calculate the amount of drug present at times \(t=1,2, \ldots, 6\) (c) What is the maximum amount of drug present at any time in this interval? At what time is this maximum amount reached? (d) Use a spreadsheet to calculate the amount of drug present in hourly intervals from \(t=0\) up to \(t=24\). (e) Show that, when \(t\) is large, the amount of drug present in the blood decreases approximately exponentially with \(t .\) Hint: Plot the values that you computed for \(a_{t}\) against \(t\) on semilogarithmic axes.
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